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The Estimation of Factors in FAVAR Models The estimation of factors in FAVAR models Erik Berggren Supervised by Professor Joakim Westerlund Master thesis at the Department of Economics May 2017 Abstract The use of factor-augmented vector autoregression (FAVAR) models has become increasingly popular in the literature of empirical macroeconomics. This paper sheds light on the different factor estimation methods that are available to researchers. More specifically, this paper examines the widely used principal component method but also the computationally simpler common correlated effects method as well as the more advanced likelihood-based method using the Gibbs sampler. The results indicate very little difference between the principal component method and the common correlated effects method, which can facilitate the estimation of FAVAR models for researchers within the field of macroeconomics. KEYWORDS: Common correlated effects, principal component analysis, Gibbs sampling, impulse response, factor models I am grateful to Joakim Westerlund for all discussions and valuable feedback during the writing of this paper. 1. Introduction Policy makers at central banks today have more than a hundred of economic indicators to monitor when conducting monetary policy. To gain insight from these variables is therefore essential for the understanding of how monetary policy affects the price level and the economy as a whole. Sims (1980) developed a tool for analysing economic time series when he introduced the vector autoregression (VAR) model as an alternative to the simultaneous equation models that had previously dominated the field of empirical macroeconomics. VAR models have become a popular choice of researchers aiming to measure the effects of monetary policy on macroeconomic variables. Bernanke and Blinder (1992) use VAR models to show that the Federal funds rate is very informative about future movements of real macroeconomic variables and Sims (1992) find that patterns in the data consistent with effective monetary policy are strikingly similar across countries. However, VAR models often require the researcher to identify only a small amount of variables that should enter the model, to avoid the curse of dimensionality. Standard VAR models do not often include more than six to eight variables in order to minimise the loss of degrees of freedom, which can be have a substantial effect if the time dimension of the dataset already is low to start with. The sparse information sets in the VAR analysis is one of the reasons for the so- called “price puzzle” that Sims (1992) noted. According to economic theory, an increase in the policy interest rate ought to lead to a decrease in the price level, but a conventional finding of studies using low-dimensional VAR models is that the price level tends to increasing following a contractionary monetary policy shock. Sims argues that the price puzzle is a result of imperfectly controlling for information that policy makers may have about future inflation. The conventional solution for this puzzle is to include a commodity price index in the model, but as Hanson (2004) argues, the inclusion of any “information variable” in a monetary VAR is often fairly ad hoc and lacks theoretical justification. The missing information in VAR models that policy makers may have at their disposal can probably not even be summarised by a single economic time series. Observable variables, such as industrial production or GDP, are imperfect measures of overall economic activity and are likely to suffer from measurement 1 errors. These insights motivated the use of factor analysis in macroeconomics as it lets the researcher use information from a large cross-section of time series without including all variables in the model. The use of factor models in time series was introduced by Sargent and Sims (1977) and Geweke (1977) and have since then been popularised by Stock and Watson (1999), who used factor models based on over a hundred series to forecast inflation. Bernanke, Boivin and Eliasz (2005) developed the factor-augmented vector autoregression (FAVAR) model that builds on earlier factor literature. Their FAVAR approach introduce two changes to the standard factor model as it allows for additional observables (in their paper it is the Federal funds rate) as well they let the unobservable factors and the observable variables jointly follow a VAR process. The use of FAVAR models in empirical macroeconomics has become increasingly popular as the availability of larger datasets increases. Belviso and Milani (2006) estimate a structural FAVAR to study shocks in the U.S. economy and Ludvigson and Ng (2009) take the FAVAR approach to study bond risk premia. In another study of monetary policy shocks, Laganá and Mountford (2005) use a U.K. dataset and conclude that FAVAR models improve results compared to VAR models without factors. The methodology by Bernanke et al. (2005) has furthermore been applied in forecasting settings and the results of Gavin and Kliesen (2008) suggest that factor-based models performed best at longer horizons, such as 12 to 24 months ahead. Matheson (2006) found that using one or two factors in the model improve results when using data from New Zealand. Though the FAVAR literature is vast, little attention has been paid to the estimation of factors in the model. The principal component (PC) approach used by Stock and Watson (2002b) is overwhelmingly favoured by researchers as Bernanke et al. (2005) conclude that the computationally more demanding likelihood-based approach led to qualitatively similar results. However, a simpler method to estimate factors is to use the cross-sectional averages of the series in the dataset, as proposed by Pesaran (2006). This method has been shown to yield consistent factor estimates and eases the estimation when the number of variables is high. The method has been extended by Kapetanios, Pesaran and Yagamata (2011), Pesaran and Tosetti (2011), and Chudnik, Pesaran and Tosetti (2011) to 2 work in different kinds of settings. To the best of my knowledge, this factor estimation approach has not been applied to FAVAR models before and the purpose of this paper is therefore to evaluate the principal component approach and the factor estimation method proposed by Pesaran (2006), called Common Correlated Effects (CCE). The use of cross-sectional averages in factor estimation for FAVAR models could lead to an even bigger surge in the use of factor models in time series analysis as it becomes even simpler to extract information from a large amount of time series without having to increase the size of the model. For the sake of comparability with Bernanke et al. (2005), this paper will also evaluate the likelihood-based estimation using the Gibbs sampler in order to investigate a more advanced method as well. The paper is organised into five sections, of which this is the introduction. Section 2 presents the econometric framework and Section 3 describes the data. The empirical results are discussed in Section 4 and Section 5 concludes. In brief, the results indicate a very small difference between the principal component approach and the common correlated effects method. The likelihood-based method is discarded due to the results being qualitatively useless. 2. Econometric framework This study follows the approach developed by Bernanke et al. (2005). They ′ ′ assume that the joint dynamics of (퐹푡 , 푌푡 ) is given by the following equation: 퐹푡 퐹푡−1 [ ] = Φ(퐿) [ ] + 휐푡 (1) 푌푡 푌푡−1 where Φ(퐿) is a conformable lag polynomial of finite order 푑 and the error term, 휐푡, is mean zero with covariance matrix Σ. The vector 푌푡 contains 푀 observable economic variables and the vector 퐹푡 represents 퐾 unobserved factors that are supposed to influence the economic variables. The factors can be thought of as unobservable concepts such as economic activity or investment climate, which cannot be represented by any observable macroeconomic series but instead several series of economic indicators. Subsequently, should the terms of Φ(퐿) that relate 푌푡 to 퐹푡−1 all be zero, then equation (1) would be reduced to a standard VAR in 푌푡. If 푌푡 in fact is related to the lagged factors then equation (1) will be referred to as a factor-augmented vector autoregression, or FAVAR. 3 The framework described above applies for all factor estimation methods but only the PC approach and the CCE method are based on equation (2) and (3) below. Equation (1) cannot be estimated directly since the factors 퐹푡 are unobservable and have to be replaced by 퐹̂푡. The estimated factors, 퐹̂푡, are assumed to be based on a number of time series that collectively are denoted by the 푁 × 1 vector 푋푡. Any given developed economy involves many different activities that can be described by various time series. The number of time series 푁 in 푋푡 is therefore assumed to be large, and may well be larger than 푇, the number of time periods. Bernanke et al. (2005) assume that the time series in 푋푡 are related to the unobservable factors 퐹푡 and the observable economic variables 푌푡 by an equation given by 푓 푦 푋푡 = Λ 퐹푡 + Λ 푌푡 + 푒푡 (2) 푓 푦 where Λ is a 푁 × 퐾 matrix of factor loadings, Λ is 푁 × 푀 and 푒푡 is a 푁 × 1 vector of error terms that are assumed to be mean zero, but may display some small degree of cross-correlation depending on the estimation method. Equation (2) express the idea that both 푌푡 and 퐹푡, which can be correlated, epitomise the common forces that drive the dynamics of the noisy measures of 푋푡. As for the estimations of 퐹̂푡, Bernanke et al. (2005) propose two different methods and it is not obvious a priori which method that should be preferred. The first method is a two-step principal component estimation and the second method is a joint estimation of equation (1) and (2) using a likelihood-based Gibbs sampling technique.
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